214
ANALYTICAL MECHANICS
(a) POINT OUTSIDE THE SPHERE. —In this case R>a. Therefore the expression for the potential may be put in the form
7=-
:-7
Therefore outside the shell the potential is the same as if the mass of the shell were concentrated at its center.
(b) POINT WITHIN THE SPHERE. —In this case R < a. Therefore
Therefore within the shell the potential is constant and equals that at the surface.
If H denotes the intensity of the field due to the shell, then
rr 67
= — 7 ~ when R >a. = 0 when R < a.
Therefore the shell attracts a particle which is outside with the same force as if all of its mass were concentrated at. its center. On the other hand the shell exerts no force on a particle which is within the shell. The distribution of 7 and H in the field are represented graphically in Fig. 108, where curve (I) represents the potential and (II) the intensity.
2. Find the expressions for the potential and the intensity due to a solid spherical mass.